The show justifies its The Main Characters Do Everything by saying that Los Angeles has far less police (9,000) for an area that is larger than New York (which has 25,000). While this figure is a bit misleading, in the Rookie-verse, it's clear our heroes are dramatically understaffed.
Nolan's status as The Heart of the group is due to the fact that he's lived a far less hardened and jaded life as a contractor rather than a career police officer. He's also from what is implied to be an extremely tight small community.
Nolan and Chen's breakup hits the latter far harder than it does the former. This makes sense because Nolan is a world-wise forty-something with a previous marriage and lots of other relationships. Its very probable that this was actually Chen's first serious relationship out of high school.
Nolan's The Ace status is something that is actually justified in-universe with the fact that while he isn't the physical equal of most trainees, his life experience actually allows him a lot more ease in dealing with people as well as simply sitting back and absorbing training. This even gets lampshaded in "Manhunt" that a lot of what they're teaching the rookies is things he learned simply by being a working man for decades.
Lucy Chen's parents being Disappointed in You seems shocking given what a The Paragon she is compared to the other officers. Then it becomes more understandable given that her parents as lifelong Los Angeles citizens would have dealt with given the LAPD's many policing scandals. They are also Bourgeois Bohemian types who tend to favor a much softer more progressive approach to social ills.
Nolan becoming a detective becomes less important due to the fact he's heavily involved in murders, terrorist attacks, and other activities anyway.
Stanton's downfall in the end isn't because he's done anything racist to civilians, he's actually undone by the fact other cops think he won't have their back.
Lots of the cases just get dropped once the protagonists are no longer involved, just like in real life. This actually gets discussed in "The Ride Along" where Nolan explains that a mentally ill young man won't get the help he needs and they can't force it. It actually makes things worse as the man's brother also gets himself involved in an armed robbery and attempted murder.
The depiction of LA is such a Wretched Hive that our protagonists are always involved in everything from mass shoot outs with armed robbers, terrorist attacks, and a variety of murders every week. While Rule of Drama applies, the implications are this is all very normal for rookie patrolmen to deal with.
Stanton's Karma Houdini status is something that not only illustrates real life difficulties in getting bad cops fired but he manages to only get his reputation destroyed in the end. He's still on the force, regardless of how his fellow cops no longer trust him.
